Circuitos Elétricos e Sistemas...

JF/CESDig & CEletro 2018/19 - Electronics Fundamentals Floyd & Buchla © Pearson Education. 28-09-2018

Circuitos Elétricos e Sistemas Digitais2018-2019 - 1.º Semestre

1JF/CESDig 2018/2019

Análise de circuitos dinâmicos (análise no tempo e em frequência)• Circuitos de corrente alternada, Formas de onda • Condensadores, Impediência e reatância • Circuitos com condensadores• Resposta em frequência de circuitos RC• Função de transferência, Diagramas de Bode• Decibel, Ponto -3 dB, Taxa de atenuação em dB por oitava e por década • Resposta temporal de circuitos RC• Circuitos RC diferenciador e integrador • Bobines/Indutâncias, Associação de indutâncias• Impediência e reatância indutiva• Circuitos RL, Resposta temporal• Resposta em frequência, Filtros RL• Circuitos RCL série e RLC paralelo, Impedância, Ressonância • Filtros passa-baixo, passa-alto, passa-banda, rejeita-banda• Frequências de corte, largura de banda e rejeita banda• Filtros, Filtros não ideais• Filtros de ordem superior (filtros de ordem n) • Taxa de atenuação por oitava e por década de filtros de ordem n


Circuitos de Corrente Alternada

2

Em corrente contínua (CC/cc), os componentes elétricos/eletrónicos passivos são caracterizados essencialmente pela respetiva resistência elétrica , e o carácter capacitivo e/ou indutivo dos componentes só se revela durante o estabelecimento do regime estacionário.

Em corrente alternada (CA/ca) o comportamento dos componentes passivos é caracterizados pela respetiva impedância Z (que é o equivalente à resistência em corrente contínua): em geral Z é uma grandeza complexa, isto é, tem uma parte real e uma parte imaginária.

A energia elétrica é-nos fornecida, em geral, na forma de sinais de tensão e corrente variáveis no tempo e cujo sentido se inverte periodicamente, percorrendo o seu ciclo de valores uma vez em cada período. A tensão/corrente alternada tem valor médio, durante um período, nulo.

A forma mais comum de corrente alternada é representada por uma função sinusoidal. Contudo, no início do século XIX a energia elétrica era fornecida quase exclusivamente fornecida na forma de corrente contínua.


Curiosidades históricas

3

A corrente contínua (CC) é correntemente gerada por dínamos , enquanto a corrente alternada (CA) é obtida a partir de alternadores . A corrente contínua apresenta algumas vantagens: as baterias podem ser usadas como sistemas de alimentação de reserva quando os dínamos falham ou em regimes de baixo consumo; os dínamos podem ser operados em paralelo de forma a aumentar a potência (o uso de alternadores em paralelo é difícil, devidos aos problemas de sincronização).

A principal vantagem da corrente alternada é a eficiência com que pode ser transmitida. A tensão alternada pode ser facilmente transformada em alta tensão, reduzindo deste modo as perdas associadas às linhas de transmissão: se a resistência da linha é e a potência transmitida é = ∙ , a perda na linha será ∙ . Assim, se a tensão transmitida for elevada e a corrente for baixa, as perdas na linha serão minimizadas.

O final do século XIX é caracterizado pela competição entre estas duas modalidades de fornecimento de energia elétrica. Vários cientistas (Thomas Edison, por exemplo) eram defensores dos sistemas de CC, mas o advento do transformador e a necessidade de transmitir energia elétrica da central até aos consumidores tornou os sistemas alternados dominantes. Nas primeiras redes de distribuição, a frequência dos sinais de CA era superior a 100 Hz (tipicamente 133 Hz).

No início do século XX, Nikola Tesla, o inventor do motor de indução, demonstrou que este não funcionaria de forma eficiente a frequências superiores a 100 Hz: nos Estados Unidos a frequência da CA é 60 Hz, enquanto que na Europa a distribuição é realizada a 50 Hz.


Formas de onda e funções sinusoidais

4

Os valores instantâneos das tensões/correntes alternadas variam no tempo, alterando periodicamente a direção da corrente/polaridade, de acordo com uma dada função denominada forma de onda. As formas de onda mais comuns são: a onda sinusoidal, a onda quadrada , a onda triangular e a onda dente de serra .

A forma de onda sinusoidal é a corrente/tensão alternada fundamental. Todas as outras formas de onda periódicas podem ser obtidas a partir da combinação de ondas sinusoidais (a onda sinusoidal fundamental e os seus harmónicos). A onda sinusoidal fica completamente caracterizada pelo período/frequência, fase, e máximos e mínimos.

Há cinco valores caraterísticos de uma forma de onda sinusoidal: o valor instantâneo (v, i), o valor de pico (VP, IP), o valor de pico-a-pico (VPP, IPP), o valor eficaz (Vef, Ief) e o valor médio (Vm, Im). Se a onda for puramente sinusoidal o valor de pico corresponde à amplitude da onda V ou I. A fase da onda é uma medida angular que especifica o valor da onda relativamente a uma referência, num dado instante de tempo.

Em corrente direta(DC)/continua (CC/cc) no regime estacionário os componentes são caracterizados apenas pela respetiva resistência elétrica, R, e o carácter capacitivo e/ou indutivo dos componentes só se revela durante o est abelecimento do regime estacionário, i.e., quando t<5τ s, onde representa chamada constante de tempo do circuito, tipicamente dada pelo produto da resistência pela capacidade, em circuitos com resistências e condensadores, ou da resistência pela indutância, no caso de circuitos com resistências e bobines.


Formas de onda/sinais comuns

5

http://www.feiradeciencias.com.br/sala15/15_07.asp

Onda sinusoidal


Forma de onda sinusoidal/Sinal sinusoidal

The sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform.

Electrical sine waves are named from the mathematical function with the same shape.

6


Sinais sinusoidais: amplitude e período

Sine waves are characterized by the amplitude and period .

The amplitude is the maximum value of a voltage or current;

the period is the time interval for one complete cycle.

The amplitude (A) of this sine wave is 20 V

The period is 50.0 ms 0 V

10 V

-10 V

15 V

-15 V

-20 V

t ( s)µ0 25 37.5 50.0

20 V

A

T

7


Período de um sinal sinusoidal

The period of a sine wave can be measured between any two corresponding points on the waveform.

T T T T

T T

By contrast, the amplitude of a sine wave is only measured from the center to the maximum point.

A

8


Frequência e período de um sinal sinusoidal

3.0 Hz

Frequency ( f ) is the number of cycles that a sine wave completes in one second.

Frequency is measured in hertz (Hz).

If 3 cycles of a wave occur in one second, the frequency is

1.0 s

9

The period and frequency are reciprocals of each other:T

f1= and

fT

1=

Thus, if you know one, you can easily find the other.

If the period is 50 ms, the frequency is 0.02 MHz = 20 kHz.

(The 1/x key on your calculator is handy for converting between f and T.)


Tensão sinusoidal: tensão de picoThere are several ways to specify the voltage of a sinusoidal voltage waveform. The amplitude of a sine wave is also called the peak value , abbreviated as VP for a voltage waveform.

0 V

10 V

-10 V

15 V

-15 V

-20 V

t ( s)µ0 25 37.5 50.0

20 V

The peak voltage of this waveform is 20 V.

VP

10


Tensão sinusoidal: tensão pico -a-picovalor eficaz

0 V

10 V

-10 V

15 V

-15 V

-20 V

t ( s)µ0 25 37.5 50.0

20 V

The voltage of a sine wave can also be specified as either the peak-to-peak or the rms value. The peak-to-peak is twice the peak value. The rms value is 0.707 times the peak value.

The peak-to-peak voltage is 40 V.

The rms voltage is 14.1 V.

VPP

Vrms

11


Tensão sinusoidal: valor médio (meio período)

0 V

10 V

-10 V

15 V

-15 V

-20 V

t ( s)µ0 25 37.5 50.0

20 V

For some purposes, the average value (actually the half-wave average) is used to specify the voltage or current. By definition, the average value is as 0.637 times the peak value.

The average value for the sinusoidal voltage is 12.7 V.

Vavg

12


Potência em circuitos resistivos

The power formulas are:

The power relationships developed for dc circuits apply to ac circuits except you must use rms values in ac circuits when calculating power.

rms rms

2

2rms

rms

P V I

VP

R

P I R

=

=

=

For example, the dc and the ac sources produce the same power to the bulb:

ac or dcsource

Bulb

0 V

0 V

220 Vdc

310 Vp

= 220 Vrms

13


Power in resistive AC circuits

Assume a sine wave with a peak value of 40 V is applied to a 100 Ω resistive load. What power is dissipated?

2 228.3 V

100 rmsV

PR

= = =Ω

Volta

ge (V

)

40

0

30

20

10

-10

-20

-30

- 40

Vrms = 0.707 x Vp = 0.707 x 40 V = 28.3 V

8 W

14


Sobreposição de tensões dc e ac

Frequently dc and ac voltages are together in a waveform. They can be added algebraically, to produce a composite waveform of an ac voltage “riding” on a dc level.

15


Exercícios1. In Europe, the frequency of ac utility voltage is 50 Hz. The period is

a. 8.3 ms b. 20.0 ms c. 60 ms d. 60 s

16

3. An example of an equation for a waveform that lags the reference is

a. v = −40 V sin (θ) b. v = 100 V sin (θ + 35o) c. v = 5.0 V sin (θ − 27o) d. v = 27 V

4. In the equation v = Vp sin θ , the letter v stands for the

a. peak value b. average value c. rms value d. instantaneous value

8. For the waveform shown, the same power would be delivered to a load with a dc voltage of

a. 21.2 V b. 37.8 V

c. 42.4 V d. 60.0 V

0 V

30 V

-30 V

45 V

-45 V

-60 V

t ( s)µ0 25 37.5 50.0

60 V



17JF/CESDig 2018/2019

Corrente alternada sinusoidal

Representação complexa e fasores


Representação de tensões sinusoidaisInstantaneous values of a wave are shown as v or i. The equation for the instantaneous voltage (v) of a sine wave is

where

If the peak voltage is 25 V, the instantaneous voltage at 50 degrees is

θsinpVv =Vp =

θ =

Peak voltage

Angle in rad or degrees

19.2 V

18

v = = 19.2 V Vp sinVp

90°

50°0°= 50°

Vp

Vp

= 25 V

A plot of the example in the previous slide (peak at 25 V) is shown. The instantaneous voltage at 50o is 19.2 V as previously calculated.


Fasor e fase inicial

00 90

90

180180 360

The sine wave can be represented as the projection of a vector rotating at a constant rate. This rotating vector is called a phasor . Phasors are useful for showing the phase relationships in ac circuits.

19

The phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference. To show that a sine wave is shifted to the left or right of this reference, a term is added to the equation given previously.

where φ = Phase shift

( )φθ ±= sinPVv


Representação Complexa e Notação Fasorial

20

É comum em corrente alternada tratar as

correntes/tensões sinusoidais, f(t)=Apcos(ωt+α),

como grandezas complexas, representando-as

usando, quer a notação exponencial, f(t)=Apej(ωt+α),

quer a notação fasorial, f=Apej(ωt+α).

O fasor f corresponde a um vetor no plano

complexo, com origem na origem do referencial,

cujo comprimento é igual à amplitude do sinal

sinusoidal (Ap), e rodado de um ângulo ωt+α (fase)

relativamente ao eixo horizontal:

f(t)=Apcos(ωt+α) f=Aej(ωt), onde A representa o complexo Apejα.

O valor instantâneo da grandeza sinusoidal é dado pela projeção do vetor no eixo

horizontal: f(t)=Apcos(ωt +a).

ωt+α

Ap

Im

Re

ω

Ο f=Apcos(ωt + α)

A “unidade SI” de ângulo plano é o radiano (rad).


Representação Complexa e Notação Fasorial

21

Uma vez que a dependência temporal da tensão e

da corrente é conhecida e é a mesma em

qualquer ponto de um circuito linear , é comum,

para simplificar a escrita, representar as

grandezas apenas pelas suas amplitudes e fase

iniciais, i.e., pelo fasor:

A=Apejα:

fasor da tensão V=Vpejα;

fasor da corrente I=Ipejφ,

Cada um destes representando:

v(t)=Vpcos(ωt+α)

e

i(t)=Ipcos(ωt+φ), respectivamente].

ωt+α

Ap

Im

Re

ω

Ο f=Apcos(ωt + α)

JF/CESDig & CEletro 2018/19 - Electronics Fundamentals Floyd & Buchla © Pearson Education. 28-09-2018 22

Corrente alternada

Formas de onda não sinusoidais


Onda quadrada

23


Pulsos ideais

Amplitude

Pulsewidth

Baseline

Amplitude

Pulsewidth

Baseline

(a) Positive-going pulse (b) Negative-going pulse

Leading (rising) edge

Trailing (falling) edge

Leading (falling) edge

Trailing (rising) edge

Ideal pulses

24


Pulsos reais

Non-ideal pulses

A0.9 A

0.1A

tr tt

fW

t t

0.5 A

A

(a) (b)Rise and fall times Pulse width

Notice that rise and fall times are measured between the 10% and 90% levels whereas pulse width is measured at the 50% level.

25


HarmónicosAll repetitive non-sinusoidal waveforms are composed of a fundamental frequency (repetition rate of the waveform) and harmonic frequencies .

Odd harmonics are frequencies that are odd multiples of the fundamental frequency.

Even harmonics are frequencies that are even multiples of the fundamental frequency.

26

A square wave is composed only of the fundamental frequency and odd harmonics (of the proper amplitude).


Onde triangular

27

https://www.quora.com/How-do-I-convert-a-triangular-wave-into-a-square-wave

Triangular waveforms have positive-going and negative-going ramps of equal duration.


Série de Fourier de uma onda triangular

28


Função dente-de-serra

29

The sawtooth waveform consists of two ramps, one of much longer duration than the other.

1T 2T 3T


Série de Fourier da função dente-de-serra

30


Gerador de funções/sinais

Function selection

Frequency

Output level (amplitude)

DC offset CMOS output

RangeAdjust

Duty cycle

Typical controls:

Outputs

Readout

Sine Square Triangle

31


Osciloscópios

HORIZONTALVERTICAL TRIGGER

5 s 5 ns

POSITION

CH 1 CH 2 EXT TRIG

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

CH 1 CH 2 BOTH

POSITION

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

SEC/DIV

POSITION

SLOPE

Ð +

LEVEL

SOURCE

CH 1

CH 2

EXT

LINE

TRIG COUP

DC AC

DISPLAY

INTENSITY

PROBE COMP5 V


5 s 5 ns

POSITION

CH 1 CH 2 EXT TRIG

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

CH 1 CH 2 BOTH

POSITION

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

SEC/DIV

POSITION

SLOPE

Ð +

LEVEL

SOURCE

CH 1

CH 2

EXT

LINE

TRIG COUP

DC AC

DISPLAY

INTENSITY

PROBE COMP5 V

Vertical


5 s 5 ns

POSITION

CH 1 CH 2 EXT TRIG

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

CH 1 CH 2 BOTH

POSITION

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

SEC/DIV

POSITION

SLOPE

Ð +

LEVEL

SOURCE

CH 1

CH 2

EXT

LINE

TRIG COUP

DC AC

DISPLAY

INTENSITY

PROBE COMP5 V

Horizontal


5 s 5 ns

POSITION

CH 1 CH 2 EXT TRIG

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

CH 1 CH 2 BOTH

POSITION

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

SEC/DIV

POSITION

SLOPE

Ð +

LEVEL

SOURCE

CH 1

CH 2

EXT

LINE

TRIG COUP

DC AC

DISPLAY

INTENSITY

PROBE COMP5 V

Trigger


5 s 5 ns

POSITION

CH 1 CH 2 EXT TRIG

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

CH 1 CH 2 BOTH

POSITION

AC-DC-GND

5 V 2 mV

VOLTS/DIV

COUPLING

SEC/DIV

POSITION

SLOPE

Ð +

LEVEL

SOURCE

CH 1

CH 2

EXT

LINE

TRIG COUP

DC AC

DISPLAY

INTENSITY

PROBE COMP5 V

Display

32



33JF/CESDig 2018/2019

Medição do desvio de fase entre duas ondas sinusoidais


Desvio de fase: atraso de fase

Volta

ge

(V)

270° 360°0° 90° 180°

40

45° 135° 225° 315°0

Angle (°)

30

20

10

-20

-30

-40

405°

Peak voltage

Reference

Notice that a lagging sine wave is below the axis at 0o

Example of a wave that lags the reference

v = 30 V sin (θ − 45o)

…and the equation has a negative phase shift

34


Desvio de fase: adianto de fase

Volta

ge

(V)

270° 360°0° 90° 180°

40

45° 135° 225° 315°0

Angle (°)

30

20

10

-20

-30

-40

Peak voltage

Reference

-45°-10

Notice that a leading sine wave is above the axis at 0o

Example of a wave that leads the reference

v = 30 V sin (θ + 45o)

…and the equation has a positive phase shift

35


Medição do desvio de faseAn oscilloscope is commonly used to measure phase angle in reactive circuits. The easiest way to measure phase angle is to set up the two signals to have the same apparent amplitude and measure the period. An example of a Multisim simulation is shown, but the technique is the same in lab.

Set up the oscilloscope so that two waves appear to have the same amplitude as shown.

Determine the period. For the wave shown, the period is

20 µs 8.0 div 160 µs

divT

= =


Medição do desvio de faseNext, spread the waves out using the SEC/DIV control in order to make an accurate measurement of the time difference between the waves. In the case illustrated, the time difference is

5 µs4.9 div 24.5 µs

divt

∆ = =

The phase shift is calculated from

24.5 µs360 360

160 µs

t

Tθ ∆ = ° = ° =

θ=55o


Medida de ângulos: radianos vs graus

Angular measurements can be made in degrees (o) or radians. The radian (rad) is the angle that is formed when the arc is equal to the radius of a circle. There are 360o or 2p radians in one complete revolution.

R

R

1.0

-1.0

0.8

-0.8

0.6

-0.6

0.4

-0.4

0.2

-0.2

00 2πππ

2π4

π4

3 π2

3π4

5 π4

7

38


Medida de ângulos: radianos vs graus

There are 2π radians in one complete revolution and 360o in one revolution. To find the number of radians, given the number of degrees:

2 radrad degrees

360

π= ×°

180deg rad

radπ°= ×To find the number of degrees, given the number of radians:

This can be simplified to: rad

rad degrees180

π= ×°

39

How many radians are in 45o?

radrad degrees

180 rad

= 45 0.785 rad180

π

π

= ×°

× ° =°

180deg rad

rad180

1.2 rad = rad

69

π

π

°= ×

°= × °

How many degrees are in 1.2 radians?


Condensadores

40


Condensadores e capacidade elétricaCapacitors are one of the fundamental passive components. In its most basic form, it is composed of two conductive plates separated by an insulating dielectric.

DieléctricoConductors

(Armaduras)

41

Capacitance is the ratio of charge to voltage

QC

V=

Rearranging, the amount of charge on a capacitor is determined by the size of the capacitor (C) and the voltage (V).

Q CV=If a 22 mF capacitor is connected to a 10 V source, the charge is 220 µC

The ability to store charge is the definition of capacitance .

An analogy:

Imagine you store rubber bands in a bottle that is nearly full.

You could store more rubber bands (like charge or Q) in a bigger bottle (capacitance or C) or if you push them in more (voltage or V). Thus, Q CV=


Capacidade elétricaA capacitor stores energy in the form of an electric field that is established by the opposite charges on the two plates. The energy of a charged capacitor is given by the equation

2

2

1CVW = where W = the energy in joules

C = the capacitance in farads

V = the voltage in volts

42

The capacitance of a parallel plate capacitor depends on three physical characteristics.

128.85 10 F/m r AC

d

ε− = ×

C is directly proportional to

and the plate area A.

the relative dielectric constant ɛr

C is inversely proportional to the distance d between the plates

d

A

ɛr


Capacidade elétrica

128.85 10 F/m r AC

d

ε− = ×

Find the capacitance of a 4.0 cm diameter sensor immersed in oil if the plates are separated by 0.25 mm.

The plate area is

The distance between the plates is

( )4.0 for oilrε =

( )( )3 2

123

4.0 1.26 10 m 8.85 10 F/m

0.25 10 mC

−−

−

× = × = ×

30.25 10 m−×

178 pF

( )2 2 3 2π 0.02 m 1.26 10 mA r π −= = = ×

43


Tipos de condensadores

Mica

MicaFoil

FoilMica

Foil

FoilMica

Foil

Mica capacitors are small with high working voltage. The working voltage is the voltage limit that cannot be exceeded.

44



Ceramic disk

Solder

Lead wire solderedto silver elec trode

Ceramicdielectric

Dipped phenolic coating

Silv er elec trodes deposited ontop and bottom of ceramic disk

Ceramic disks are small nonpolarized capacitors They have relatively high capacitance due to high εr.

45



Plastic Film

Lead wire

High-purityfoil electrodes

Plastic filmdielec tric

Outer wrap ofpolyester film

Capacitor section(alternate strips offilm dielectric andfoil electrodes)

Solder coated end

Plastic film capacitors are small and nonpolarized. They have relatively high capacitance due to larger plate area.

46



Electrolytic (two types)

Symbol for any electrolytic capacitor

Al electrolytic

+

_

Ta electrolytic

Electrolytic capacitors have very high capacitance but they are not as precise as other types and tend to have more leakage current. Electrolytic types are polarized.

47


Condensadores de capacidade variável

VariableVariable capacitors typically have small capacitance values and are usually adjusted manually.

A solid-state device that is used as a variable capacitor is the varactor diode; it is adjusted with an electrical signal.

48


Leitura do valor da capacidade

Capacitors use several labeling methods. Small capacitors values are frequently stamped on them such as .001 or .01, which have implied units of microfarads.

+++

+

VT

TV

TT

47 MF

.022

Electrolytic capacitors have larger values, so are read as mF. The unit is usually stamped as mF, but some older ones may be shown as MF or MMF).

49

A label such as 103 or 104 is read as 10x103 (10,000 pF) or 10x104 (100,000 pF) respectively. (Third digit is the multiplier.)

When values are marked as 330 or 6800, the units are picofarad.

What is the value of each capacitor?

Both are 2200 pF. 222 2200


Associação de condensadores

50


Associação de condensadores em série

When capacitors are connected in series, the total capacitance is smaller than the smallest one. The general equation for capacitors in series is:

T

1 2 3 T

11 1 1 1

...C

C C C C

=+ + + +

The total capacitance of two capacitors is T

1 2

11 1

C

C C

=+…or you can use the product-over-sum rule

51

If a 0.001 mF capacitor is connected in series with an 800 pF capacitor, the total capacitance is 444 pF

0.001 µF 800 pF

C1 C2

0.001 µF 800 pF

C1 C2

444 pF

CTi

1 2 = + ; = = =

→

= +

→ = +


Associação de condensadores em paralelo

When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitors. The general equation for capacitors in parallel is

T 1 2 3 ... nC C C C C= + + +

1800 pF

If a 0.001 mF capacitor is connected in parallel with an 800 pF capacitor, the total capacitance is

0.001 µF 800 pF

C1 C2

52

0.001 µF 800 pF

C1 C2CT

1800 pF

ii2i1v

= + ; = + =

→ = +


Curvas de carga e de descarga de condensadores

53


Carga e descarga de um condensador

Dielectric

Plates

Leads

Electrons

BA

−

−

−−

+

+

+

+

−

−

+

+

+

+

−

Initially uncharged

+ −BA

−−

−−

−−

−

+

+

+

−−

−

−

− − − − − −

−

−−−

+

+

+

+

Charging

+ −BA

VS

+

+

+++++++++

−

−

−−−−−−−−−

Fully charged

BA

VS

−+

−+

−+−+−+−+−+−+−+−+−+

Source removed

The charging process…

A capacitor with stored charge can act as a temporary battery.

54

VS


Condensador descarregado

Dielectric

Plates

Leads

Electrons

BA

−

−

−−

+

+

+

+

−

−

+

+

+

+

−

Initially uncharged


55


Condensador a carregar

+ −BA

−−

−−

−−

−

+

+

+

−−

−

−

− − − − − −

−

−−−

+

+

+

+

Charging


56

C

vci = → () =

() = d()d ; = 1

!"

VS

vC


Constante de tempo RC de um circuitoWhen a capacitor is charged through a series resistor and dc source, the charging curve is exponential.

C

R

Iinitial

t0(b) Charging current

Vfinal

t0(a) Capacitor charging voltage

57

The general voltage formula is

v =VF + (Vi − VF) e−t/RC

VF = final value of voltageVi = initial value of voltagev = instantaneous value of voltage

The final capacitor voltage is greater than the initial voltage when the capacitor is charging.


Condensador carregado

58

C

vci = = # → =

= d d = 0

Após carregar, i.e., quando = #, deixa de haver corrente no circuito. Em CC o

condensador comporta-se, após o período de transiente, como um aberto.

R

vC=

+ −BA

VS

+

+

+++++++++

−

−

−−−−−−−−−

Fully charged

VS



Condensador carregado

BA

VS

−+

−+

−+−+−+−+−+−+−+−+−+

Source removed


A capacitor with stored charge can act as a temporary battery. The energy stored is given by:

59

C

vc

& = = → ' = 12(()


Descarga de um condensadorWhen a charged capacitor is discharged through a resistor, the discharge curve is also an exponential. (Note that the current is negative.)

t

t

−Iinitial

0

(b) Discharging current

Vinitial

0(a) Capacitor discharging voltage

C

R

60



VF = final value of voltageVi = initial value of voltagev = instantaneous value of voltage

The final capacitor voltage is less than the initial voltage when it is discharging.

+VS-


Resposta temporal de circuitos dinâmicos

61

Consideremos o circuito RC sujeito a uma tensão em degrau:

)*(t)=a·H(t)= + !" + ()

Considerando o caso = 0 para , 0, temos para . 0s:

)*(t)=a·H(t)=

+

0 1

Derivando ambos os termos da equação (1) obtém-se 0

1

(2)

Assumindo uma solução do tipo 234 e substituindo na equação (2), tem-se 0

234 2534, de onde se conclui que 5 61/.

Se no instante inicial a tensão aos terminais do condensador for nula, temos que 0

8

9e :;< = >3!/?, com @ .

Trocando a ordem dos componentes, obtém-se :;< A > 1 6 3!BC

C


Universal exponential curves

Specific values for current and voltage can be read from a universal curve. For an RC circuit, the time constant is

τ RC=100%

80%

60%

40%

20%

00 1τ 2τ 3τ 4τ 5τ

99%98%

95%

86%

63%

37%

14%

5% 2% 1%

Number of time constants

Per

cent

of f

inal

val

ue

Rising exponential

Falling exponential

62



VF = final value of voltageVi = initial value of voltagev = instantaneous value of

voltage

The final capacitor voltage is less than the initial voltage when it is discharging.


Universal exponential curvesThe universal curves can be applied to general formulas for the voltage (or

current) curves for RC circuits. The general voltage formula is


VF = final value of voltage

Vi = initial value of voltage

v = instantaneous value of voltage

The final capacitor voltage is greater than the initial voltage when the capacitor is

charging, or less that the initial voltage when it is discharging.

63

τ RC=

The general voltage formula gives that when t = τ s:

v (t=τ=RC)=VF + (Vi − VF) e−1≈ VF + (Vi− VF)×0,63. Se Vi=0 V, têm-

se v (t)=VF (1− e−t/RC). v (t=τ=RC)=VF (1− e−1) ≈ VF ×0,63.


Impedância em circuitos de corrente alternada sinusoidal


Circuitos de corrente alternada: impedância

65

Em corrente alternada (ca) sinusoidal os componentes passivos são caracterizados por uma grandeza complexa designada impedância , Z. A impedância representa a oposição do componente ao estabelecimento de uma corrente sinusoidal.

Em corrente alternada sinusoidal a lei de Ohm toma a forma: V=Z·I, onde V e I representam a amplitude/valor eficaz da tensão e da amplitude da corrente, respetivamente, e Z é a impedância do componente.

Em CA a lei de Ohm não é válida para os valores ins tantâneos das grandezas. Para assegurar a sua correta aplicação, a tensão e a cor rente devem ser expressas de forma consistente, i.e., ambas referidas aos valore s de pico, aos valores eficaz, etc.

Em geral, a impedância Z é uma grandeza complexa. Na representação complexa temos:

Z=V.ejωt/I.ejωt = Vpejφ/Ipejαj=|Z|ejq,

onde θ=φ-α representa a diferença de fase entre a tensão e a corrente. Na representação algébrica Z toma a forma: Z=R+jX, j2=-1, onde R e X representam as partes resistiva e reativa da impedância do elemento ou da parte do circuito em análise: Rdesigna-se por resistência óhmica (unidade SI: ohm Ω) e X por reactância (unidade SI: ohm Ω). Se X for negativo diz-se que a reactância é capacitiva X=XC; se X for positivo diz-se que a reactância é indutiva X=XL. O modulo e a fase de Z são dados, respetivamente, por: D = + E , F = tan! E/


Impedância

66

Na representação complexa temos: Z=V.ejωt/I .ejωt = Vpejφ/Ipejϕ=|Z|ejθ, onde θ=φ−ϕ

representa a diferença de fase entre a tensão e a corrente.

Na representação algébricaZ toma a forma:Z=R+jX, j2=-1, ondeR e X representam as

partesresistiva e reactiva da impedância do elemento ou parte do circuito em análise.R

designa-se porresistência óhmica(uSI: ohmΩ) eX por reactância(uSI: ohmΩ).

SeX for negativo diz-se que areactânciaé capacitivaX=XC; seX for positivo diz-se que

a reactânciaé indutivaX=XL.

O módulo e a fase deZ são dados, respectivamente, por:D = + E eF = tan! I9 .

O inverso da impedância designa-seadmitância complexaou simplesmenteadmitância

Y: Y=1/Z=G+jB.

G denomina-secondutância(unidade SI: S) e Bsusceptância(unidade SI: S).

Associação de impedâncias em série:Zeq=Z1+ Z2+ … + Zn-1+ Zn.

Associação de impedâncias em paralelo:1/Zeq= 1/Z1+ 1/Z2+ … + 1/Zn-1+ 1/Zn.


A impedância é uma grandeza complexa, que pode ser escrita como:

67

Impedância de condensadores

= D = D ∙ D =

Z = + KEX corresponde à componente imaginária da impedância, designada reactância.

D = 1K2LM

6KE , ondeE = 12LM

1P

Lei de Ohm em corrente alternada:

Seja uma tensão alternada sinusoidal QR = 0cos P . Tendo presente que

A = TU , A = −P0sin P = P0cos(P + 90).Na representação complexa: fazendo QR = 03XY, temos:

A = 03XY = KP03XY. Pela lei de Ohm = Z[ obtêm-se: Z=1/(KP)=−KE.

Impedância do condensador:

R representa a parte real (resistiva) da impedância


Em corrente alternada os componentes passivos são caraterizados pela respetiva impedância Z, e a lei de Ohm toma a forma:

Because I is the same everywhere in a series circuit , you can obtain the voltages across different components by multiplying the impedance of that component by the current. As consequence the impedance of the series is equal to the sum of the individual impedances.

68

Associação de impedâncias

= D ; D ∙ ; D

Z1 Z2 Z…

Zn

Zeq=Z1+Z2…+Zn

ZnZ1 Z2

Because V is the same everywhere in a parallel circuit , you can obtain the current across each component by dividing the voltage across it by its impedance. As consequence the reciprocal of the impedance of the parallel is equal to the sum of the individual impedance reciprocals.

Associação em série de impedâncias

Associação em paralelo de impedâncias

[\]

[+

[+ …+

[U


Impedância de condensadores

69


Impedância de um condensadorWhen a sine wave is applied to a capacitor, there is a phase shift between voltage and current such that current always leads the voltage by 90o.

VC0

I 0

90o

70

V V

V IZ I ZZ I

= = = DA = 1

K2LM

1

KP K 6

1

P→ KP _ 1`

1`3XY→

KP→1`

XY


Reactância capacitiva

Capacitive reactance is the opposition to ac by a capacitor. The equation for capacitive reactance is

D = XY = −K

XY = −KE →E =

abThe reactance of a 0.047 uF capacitor when a sinusoidal voltage of frequency of 15 kHz is applied is 226 Ω.

When capacitors are in series, the total reactance is the sum of the individual reactances. That is,

Assume three 0.033 uF capacitors are in series with a 2.5 kHz ac source. What is the total reactance? The reactance of each capacitor is

5.79 kΩ

C( ) C1 C2 C3 Ctot nX X X X X= + + + ⋅⋅⋅ +

( )( )1 1

1.93 k2π 2π 2.5 kHz 0.033 µFCX

fC= = = Ω

C( ) C1 C2 C3

1.93 k 1.93 k 1.93 ktotX X X X= + +

= Ω + Ω + Ω =71


Reactância capacitiva

When capacitors are in parallel, the total reactance is the reciprocal of the sum of the reciprocals of the individual reactances. That is,

If the three 0.033 uF capacitors from the last example are placed in parallel with the 2.5 kHz ac source, what is the total reactance?

643 Ω

C( )

C1 C2 C3 C

11 1 1 1tot

n

X

X X X X

=+ + + ⋅⋅⋅+

The reactance of each capacitor is 1.93 kΩ

C( )

C1 C2 C3

1 11 1 1 1 1 1

+ +1.93 k 1.93 k 1.93 k

totX

X X X

= = =+ +

Ω Ω Ω

72


Circuitos com condensadores

73


Desfasamento capacitivo (Capacitive phase shift)

When a sine wave is applied to a capacitor, there is a phase shift between voltage and current such that current always leads the voltage by 90o.

VC0

I 0

90o

74

V V

V IZ I ZZ I

= = = DA = 1

K2LM

1

KP→ KP _


Divisor de tensão com condensadoresTwo capacitors in series are commonly used as a capacitive voltage divider. The

capacitors split the output voltage in proportion to their reactance (and inversely

proportional to their capacitance).

What is the output voltage for the capacitive voltage divider?

Vout

( )( )11

1 14.82 k

2π 2π 33 kHz 1000 pFCXfC

= = = Ω

( )( )22

1 1482

2π 2π 33 kHz 0.01 µFCXfC

= = = Ω

1000 pF

0.01 µFC2

C1

1.0 Vf = 33 kHz

2

( )

482 1.0 V =

5.30 kC

out sC tot

XV V

X

Ω = = Ω

( ) 1 2

4.82 k 482 5.30 kC tot C CX X X= +

= Ω + Ω = Ω

91 mV

75


Divisor de tensão com condensadoresInstead of using a ratio of reactances in the capacitor voltage divider

equation, you can use a ratio of the total series capacitance to the output

capacitance (multiplied by the input voltage). The result is the same. For the

problem presented in the last slide,

Vout

( )( )1 2( )

1 2

1000 pF 0.01 µF909 pF

1000 pF 0.01 µFtot

C CC

C C= = =

+ +1000 pF

0.01 µFC2

C1

1.0 Vf = 33 kHz

( )

2

909 pF1.0 V =

0.01 µFtot

out s

CV V

C

= =

91 mV

76


Circuito série com condensador e resistência

77

R

VR

C

VR leads VS VC lags VS

I leads VS

I

VS

VC

Representa a impedância no plano complexo para o circuito acima, com R = 1.2 kΩ e XC = 960 Ω, e determine a fase da impedância.

When both resistance and capacitance are in a series circuit, the phase angle between the applied voltage and total current is between 0° and 90°, depending on the values of resistance and reactance.

( ) ( )2 21.2 k + 0.96 k

1.33 k

Z = Ω Ω

= Ω

1 0.96 ktan

1.2 k39

θ − Ω=Ω

= °


Phasor diagrams that have reactance phasors can only be drawn for a single

frequency because X is a function of frequency.

As frequency changes, the impedance

triangle for an RC circuit changes as

illustrated here because XC decreases with

increasing f. This determines the frequency

response of RC circuits.

θ

Z3

XC1

XC2

XC3

Z2

Z1

12

3

1

2

f

f

f

3

Increasing fθ

θ

R

78

Variação do ângulo de fase com a frequência

Assume the current in the previous example is 10 mArms. Sketch the voltage

phasor diagram. The impedance triangle from the previous example is shown for

reference.

The voltage phasor diagram can be found from Ohm’s law. Multiply each

impedance phasor by 10 mA.


Circuitos RC paralelo

79

In a parallel RC circuit, the admittance phasor is the sum of the conductance and capacitive susceptance phasors. The magnitude can be expressed as

2 2 + CY G B=From the diagram, the phase angle is

1tan CB

Gθ − =

VS G BC

For parallel circuits, it is useful to introduce two new quantities (susceptance and admittance) and to review conductance.

Conductance is the reciprocal of resistance. 1

GR

=

Admittance is the reciprocal of impedance.

Capacitive susceptance is the reciprocal of capacitive reactance.

1C

C

BX

=1

YZ

=

2CB fCπ=


Circuitos RC paralelo

80

VS C

0.01 µFf = 10 kHz

R

1.0 kΩ

Desenhe o diagrama do fasor da admitância para os circuito acima

1 11.0 mS

1.0 kG

R= = =

Ω( )( )2 10 kHz 0.01 F 0.628 mSCB π µ= =

As magnitudes da condutância e da susceptância são:

( ) ( )2 22 2 + 1.0 mS + 0.628 mS 1.18 mSCY G B= = =

If the voltage in the previous example is 10 V, sketch the current phasor diagram. The admittance diagram from the previous example is shown for reference.

The current phasor diagram can be found from Ohm’s law. Multiply each admittance phasor by 10 V.


Circuitos RC Séries-Paralelo

Series-parallel RC circuits are combinations of both series and parallel elements. These circuits can be solved by methods from series and parallel circuits.

For example, the components in the green box are in series:

The components in the yellow box are in parallel: 2 2

2 2 22 2

C

C

R XZ

R X=

+

R1 C1

R2 C2

Z1Z2

The total impedance can be found by converting the parallel components to an equivalent series combination, then adding the result to R1 and XC1 to get the total reactance.

2 21 1 1CZ R X= +


Resposta de circuitos RC a sinais sinusoidais

82


When both resistance and capacitance are in a series circuit, the phase angle between the applied voltage and total current is between 0° and 90°, depending on the values of resistance and reactance.

R

VR

C

VR leads VS VC lags VS

I leads VS

I

VS

VC

83

Resposta de circuitos RC a sinais sinusoidais(Sinusoidal response of RC circuits)


θ

(phase lag)

φ

φ

(phase lag)

V

RVR

Vout

C VoutVin

Vin

Vout

Vin

For a given frequency, a series RC circuit can be used to produce a phase lag by a specific amount between an input voltage and an output by taking the output across the capacitor. This circuit is also a basic low-pass filter, a circuit that passes low frequencies and rejects all others.

84

Análise de circuitos RC série

1tan CX

Rθ − =

A diferença de fase φ entre as tensões Vout e Vin é φ=90-θ, onde θ=tan-1(Xc/R):

E = πb =

ω

Ohm’s law is applied to series RC circuits using Z, V, and I. V V

V IZ I ZZ I

= = =

Because I is the same everywhere in a series circuit, you can obtain the voltages across different components by multiplying the impedance of that component by the current as shown in the following example.


θ

θ

(phase lead)

(phase lead)

V

R

VC

VoutC

VoutVin

Vin

Vout

Vin

Reversing the components in the previous circuit produces a circuit that is a basic lead network. This circuit is also a basic high-pass filter, a circuit that passes high frequencies and rejects all others. This filter passes high frequencies down to a frequency called the cutoff frequency.

85

Circuitos CR série

A diferença de fase φ entre as duas tensões é φ=θ=tan-1(Xc/R), com

E = πb =

ω

1tan CX

Rθ − =


Medição da diferença de faseAn oscilloscope is commonly used to measure phase angle in reactive circuits. The easiest way to measure phase angle is to set up the two signals to have the same apparent amplitude and measure the period. An example of a Multisim simulation is shown, but the technique is the same in lab.

Set up the oscilloscope so that two waves appear to have the same amplitude as shown.

Determine the period. For the wave shown, the period is

20 µs 8.0 div 160 µs

divT

= =


Medição da diferença de fase

Next, spread the waves out using the SEC/DIV control in order to make an accurate measurement of the time difference between the waves. In the case illustrated, the time difference is

5 µs4.9 div 24.5 µs

divt

∆ = =

The phase shift is calculated from

24.5 µs360 360

160 µs

t

Tθ ∆ = ° = ° =

θ=55o


Exercícios1. The capacitance of a capacitor will be larger if

a. the spacing between the plates is increased. b. air replaces oil as the dielectric.

c. the area of the plates is increased. d. all of the above.

88

2. The major advantage of a mica capacitor over other types is

a. they have the largest available capacitances. b. their voltage rating is very high

c. they are polarized. d. all of the above.

3. Electrolytic capacitors are useful in applications where

a. a precise value of capacitance is required. b. low leakage current is required.

c. large capacitance is required. d. all of the above.

4. If a 0.015 µF capacitor is in series with a 6800 pF capacitor, the total capacitance is

a. 1568 pF. b. 4678 pF. c. 6815 pF. d. 0.022 µF.


Exercícios5. Two capacitors that are initially uncharged are connected in series with a dc source. Compared to the larger capacitor, the smaller capacitor will have

a. the same charge. b. more charge. c. less voltage. d. the same voltage.

89

6. When a capacitor is connected through a resistor to a dc voltage source, the charge on the capacitor will reach 50% of its final charge in

a. less than one time constant. b. exactly one time constant.

c. greater than one time constant. d. answer depends on the amount of voltage.

7. When a capacitor is connected through a series resistor and switch to a dc voltage source, the voltage across the resistor after the switch is closed has the shape of

a. a straight line. b. a rising exponential. c. a falling exponential. d. none of the above.

8. The capacitive reactance of a 100 µF capacitor to 60 Hz is

a. 6.14 kΩ. b. 265 Ω. c. 37.7 Ω. d. 26.5 Ω

9. If an sine wave from a function generator is applied to a capacitor, the current will

a. lag voltage by 90o. b. lag voltage by 45o. c. be in phase with the voltage.

d. none of the above.



91JF/CESDig 2018/2019

Análise de circuitos dinâmicos (análise no tempo e em frequência)

• Resposta em frequência de circuitos RC

• Função de transferência

• Decibel

• Ponto -3 dB

• Diagramas de Bode

• Frequências de corte, largura de banda e rejeita banda

• Resposta temporal de circuitos RC

• Circuitos RC diferenciador e integrador


Impedância em circuitos de corrente alternada sinusoidal


A impedância depende da frequência

93

Em corrente alternada (CA) sinusoidal os componentes passivos são caracterizados por uma grandeza complexa designada impedância , Z. A impedância representa a oposição do componente ao estabelecimento de uma corrente sinusoidal, e em geral depende da frequência do sinal sinusoidal. A lei de Ohm toma a forma: V=Z·I, onde V e I representam a amplitude/valor eficaz da tensão e da amplitude da corrente, respetivamente, e Z é a impedância do componente (f unção da frequência do sinal sinusoidal). Em CA a lei de Ohm não é válida para o s valores instantâneos das grandezas. Para assegurar a sua correta aplicação, a tensão e a corrente devem ser expressas de forma consistente, i.e., ambas referid as aos valores de pico, aos valores eficaz, etc.

Em geral, a impedância Z é uma grandeza complexa. Na representação complexa temos:

Z=V(ω).ejωt/I(ω)ejωt=|Z(ω)|ejq, |D(ω)| = + E(ω) , F = tan! E(ω)/

onde θ=φ-α representa a diferença de fase entre a tensão e a corrente. Na representação algébrica Z toma a forma: Z(ω)=R+jX(ω), j2=-1, onde R e X(ω) representam as partes resistiva e reativa da impedância do elemento ou da parte do circuito em análise: Rdesigna-se por resistência óhmica (unidade SI: ohm Ω), e aqui considera-se que não depende da frequência do sinal sinusoidal , e X é reactância (unidade SI: ohm Ω), e é função da frequência do sinal sinusoidal. Se X for negativo diz-se que a reactância é capacitiva X=XC; se X for positivo diz-se que a reactância é indutiva X=XL.


Função de transferência do circuito, Diagramas de

Bode, frequências de corte e largura de banda


Resposta em Frequência de um Circuito(Ver Microelectronics Circuits, A. S. Sedra & K. C. Smith, Saunders College Publishing, capítulo 1)

Considere o circuito linear abaixo, ao qual é aplicada uma tensãovin(t)=VINcos(ωt), representada noesquema pela amplitudeVIN(ω).

VOUT

IIN

ZVIN(ω)

IOUT

Pretende-se estudar o comportamento do sinal de saída,VOUT, em função da frequência do sinal de

entradaVIN(ω), i.e., caracterizar a resposta em frequência do circuito.

A resposta em frequênciaé descrita pelafunção de transferência do circuito, H(ω), que é a razão

entre a tensão a saída,VOUT(ω), e a tensão aa entrada,VIN(ω), com a saída em aberto (i.e.,IOUT=0).

Em geral,H(ω) é uma função complexa:

H(ω)=|H(ω)|ejθ(ω) ,

onde |H(ω)|=|VOUT/V IN| e θ(ω) é a ddf entre a tensão a entrada e a tensão a saída.


Diagramas de Bode

Considere o circuito linear abaixo, ao qual é aplicada uma tensãovin(t)=VINcos(ωt), representada noesquema pela amplitudeVIN(ω):

VOUT

IIN

ZVIN(ω)

IOUT

θ(ω,f)

θ+

θ-

20 defg(P, M)

Log (ω,f)

Conhecida a resposta em frequência, descrita pelafunção de transferência do circuito, H(ω), que é

a razão entre a tensão a saída,VOUT(ω), e a tensão aa entrada,VIN(ω), com a saída em aberto (i.e.,

IOUT=0), H(ω)=|H(ω)|ejθ(ω)=H(ω)ejθ(ω), é corrente apresentar a resposta em frequência do circuito

representando graficamente o comportamento do móduloH(ω) e da faseθ(ω) da função de

transferência, obtendo-se osdiagramas de Bode, representações gráficas das funçõesH(ω) e θ(ω) na

forma20 Logg(P, M)eθ P = argl(P, M) , enoeixoω/M é representado o logaritmo deω/M:

(Ver Microelectronics Circuits, A. S. Sedra & K. C. Smith, Saunders College Publishing, capítulo 1)

Log (ω,f)


Diagramas de Bode

θ(ω,f)

θ+

θ-

20 Log |l P, M |

Log (ω,f)Log (ω,f)

Diagramas de Bode: representações gráficas das funções |H(ω)| e θ(ω) na forma

20 Log |l P, M |e θ P = argl(P, M), e o eixo das abcissas corresponde ao

logaritmo deω/M:

)707.0(2/1 ≈

=

IN

OUT

V

Vlog20dB

Decibel: -3 dB

( )( )

( )( ) 2

1

ωV

ωV

ωV

ωVdB3

csIN

csOUT

ciIN

ciOUT ==≡-

Ver: https://pt.wikipedia.org/wiki/Decibel

https://en.wikipedia.org/wiki/Decibel


Decibéis

Circuit/filter responses are often given in terms of decibels, which is defined as

dB 10log out

in

P

P

=

dB 20log out

in

V

V

=

Another useful definition for the decibel, when measuring voltages across the same impedance is

Because it is a ratio, the decibel is dimensionless. One of the most important decibel ratios occurs when the power ratio is 1:2. This is called the −3 dB frequency, because

1dB 10log 3 dB

2 = = −


DecibelThe decibel (symbol: dB) is a unit of measurement used to express the ratio of one value of a physical property to another on a logarithmic scale. It can be used to express a change in value (e.g., +1 dB or −1 dB) or an absolute value. In the latter case, it expresses the ratio of a value to a reference value; when used in this way, the decibel symbol should be appended with a suffix that indicates the reference value, or some other property. For example, if the reference value is 1 volt, then the suffix is "V" (e.g., "20 dBV"), and if the reference value is one milliwatt, then the suffix is "m" (e.g., "20 dBm").[1]

Uses: ElectronicsIn electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, decibels denote signal gain or loss from a transmitter to a receiver through some medium (free space, waveguide, coaxial cable, fiber optics, etc.) using a link budget.

The decibel unit can also be combined with a suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". A power level of 0 dBmcorresponds to one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW). In professional audio specifications, a popular unit is the dBu. This is relative to the root mean square voltage which delivers 1 mW (0 dBm) into a 600-ohm resistor, or √1 mW×600 Ω ≈ 0.775 VRMS. When used in a 600-ohm circuit (historically, the standard reference impedance in telephone circuits), dBu and dBm are identical.


dB Power ratio Amplitude ratio 100 10000000000 10000090 1000000000 3162380 100000000 1000070 10000000 316260 1000000 100050 100000 316.2 40 10000 10030 1000 31.62 20 100 1010 10 3.162 6 3.981 ≈ 4 1.995 ≈ 2 3 1.995 ≈ 2 1.413 ≈ √21 1.259 1.122 0 1 1−1 0.794 0.891 −3 0.501 ≈ 1⁄2 0.708 ≈ √ 1⁄2−6 0.251 ≈ 1⁄4 0.501 ≈ 1⁄2−10 0.1 0.3162−20 0.01 0.1 −30 0.001 0.03162−40 0.0001 0.01 −50 0.00001 0.003162−60 0.000001 0.001 −70 0.0000001 0.0003162−80 0.00000001 0.0001−90 0.000000001 0.00003162−100 0.0000000001 0.00001

An example scale showing power ratios x, amplitude ratios √x, and dB equivalents 10 log10 x.


Frequências de corte e largura de banda

Função de transferência de um circuito, H(ω), definida como a razão entre a tensão à saída, VOUT(ω), e atensão à entrada, VIN(ω), com a saída em aberto (IOUT=0): H(ω)=|H(ω)|ejθ(ω) , onde |H(ω)|=|VOUT/VIN| e θ(ω)

é a ddf entre a tensão à entrada e a tensão à saída.

VOUT

IIN

ZVIN(ω)

IOUT H(ω,f)

ω,f(ω,f )ci (ω,f )cs

Pontos 3 dB

Largura de banda

1,00,7

θ(ω ,f)

ω,f

θ+

θ-

Define-selargura de banda de um circuito,LB, como ogama de frequências, ∆f, para a qual o

módulo da função de transferência é maior ou igual a1/√2, ver gráficoH(ω,f). (Ter presente que

f=ω/2π.) Quandofci=0, diz-se que o circuito é umpassa-baixo; sefcs=∞, o circuito funciona como um

passa-alto. Se 0<fci<fcs<∞, o circuito actua comopassa-banda, permitindo apenas a passagem de

sinais de frequênciaf na banda [fcs, fci]. Há ainda circuitos cuja resposta em frequência pode ser

representada como a combinação de um passa-alto (pa) com um passa-baixo (pb), em quefc-pb<fc-pa:

circuitosrejeita-banda. Estes não permitem a passagem de sinais de frequênciaf∈[fc-pb, fc-pa].

fc(i,s): frequência de corte, c, (i: inferior; s: superior)

)707.0(2/1 ≈

(Ver Microelectronics Circuits, A. S. Sedra & K. C. Smith, Saunders College Publishing, capítulo 1)


Frequências de corte e largura de banda

VOUT

IIN

ZVIN(ω)

IOUT

H(ω,f)

ω,f(ω,f )ci (ω,f )cs

Pontos 3 dB

Largura de banda

1,00,7

fc(i,s): frequência de corte, c, (i: inferior; s: superior)

)707.0(2/1 ≈

Define-selargura de banda de um circuito,LB, como ogama de frequências, ∆f, para a

qual o módulo da função de transferência é maior ou igual a1/√2, ver gráficoH(ω,f). (Ter

presente quef=ω/2π.) Quandofci=0, diz-se que o circuito é umpassa-baixo; se fcs=∞, o

circuito funciona como umpassa-alto. Se 0<fci<fcs<∞, o circuito actua comopassa-banda,

permitindo apenas a passagem de sinais de frequênciaf na banda [fcs, fci]. Há ainda circuitos

cuja resposta em frequência pode ser representada como a combinação de um passa-alto

(pa) com um passa-baixo (pb), em quefc-pb<fc-pa: circuitos rejeita-banda. Estes não

permitem a passagem de sinais de frequênciaf∈[fc-pb, fc-pa].


Resposta em frequência de circuitos com condensadores

103


Resposta em frequência de circuitos RC

104

Consider a signal is applied to an RC circuit, and the output is taken across the capacitor as shown:



10 V dc

VoutV in

100 Ω1 Fµ

10 V dc

0

10 V dc

0

When a signal is applied to an RC circuit, and the output is taken across the capacitor as shown, the circuit acts as a low-pass filter.

As the frequency increases, the output amplitude decreases.



10 V dc

VoutV in

100 Ω1 Fµ

10 V dc

0

10 V dc

0



1ƒ = 1 kHz

8.46 V rms10 V rms Ω100 Fµ



10 V dc

VoutV in

100 Ω1 Fµ

10 V dc

0

10 V dc

0



1ƒ = 1 kHz


1.57 V rms

10 V rms

1ƒ = 10 kHzΩ100

Fµ



10 V dc

VoutV in

100 Ω1 Fµ

10 V dc

0

10 V dc

0



1ƒ = 1 kHz


1.57 V rms

10 V rms

1ƒ = 10 kHzΩ100

Fµ0.79 V rms

10 V rms

1ƒ = 20 kHzΩ100

Fµ



10 V dc

VoutV in

100 Ω1 Fµ

10 V dc

0

10 V dc

0



Plotting the frequency response: Vout (V)

9.98

8.46

1.570.79

0.1 1 10 20 100f (kHz)

9

8

7

6

5

4

3

2

1

1ƒ = 1 kHz


1.57 V rms

10 V rms

1ƒ = 10 kHzΩ100

Fµ0.79 V rms

10 V rms

1ƒ = 20 kHzΩ100

Fµ


Frequência de corte

Cutoff frequency (fc): The frequency at which the output voltage of a filter is 70.7% (1/√2) of the maximum output voltage, or the frequency at which the output power is reduced to half it its maximum valu e.

Frequency response: In electric circuits, the variation of the output voltage (or current) over a specified range of frequencies.

Vout (V)

9.98

8.46

1.570.79

0.1 1 10 20 100f (kHz)

9

8

7

6

5

4

3

2

1

fc

10/√2)

fc~1592 Hz


Função de Transferência de Circuitos RC: filtro RC passa-baixo

Circuito RC Passa-Baixo

)(tan

)(1

1)(H)(

ZZ

Z

V

V)(H)(H

1

2

IN

OUTj

RC

RCH

eCR

C

ω−=θ

ω+=ω=ω

+==ω=ω

−

θ

A amplitude da tensão aos terminais do condensador (tensão de saída VOUT), decresce à medida que a

frequência do sinal de entrada, VIN, aumenta.

A frequência angular de corte deste circuito,ωc, é |H(ωc)|=|VOUT/VIN|=1/√2: ω=ωc=1/RC. A

frequência angularωci=0 eωcs=1/RC. A largura de banda éLB=ωcs/2π - ωci/2π =1/(2πRC).

O circuito comporta-se como umfiltro passa-baixo: só os sinais de entrada de frequência inferior a

ωc são “bem” transferidos para a saída.

i

CVIN VOUT

R

IOUT=0


g P = pq1` =1

1 KP

rsst

3!Xuvw<uR

sst

P

9frequência de corte

Região transparente

Região opaca

Diagrama de Bode - circuito RC passa-baixo

K=1

Microelectronic_Circuits_6th_Edition_Sedra


Resposta em frequência de circuitos “CR”

113

Considered the RC circuit obtained by reversing the components, and taking the output across the resistor as shown:



Vin

10 V dc

0

Vout

0 V dc10 V dc 100 Ω1 Fµ

Reversing the components, and taking the output across the resistor as shown, the circuit acts as a high-pass filter.

As the frequency increases, the output amplitude also increases.



Vin

10 V dc

0

Vout

0 V dc10 V dc 100 Ω1 Fµ



ƒ = 100 Hz

0.63 V rms10 V rms

100 Ω1 Fµ



Vin

10 V dc

0

Vout

0 V dc10 V dc 100 Ω1 Fµ



ƒ = 100 Hz

0.63 V rms10 V rms

100 Ω1 Fµ

ƒ = 1 kHz

5.32 V rms10 V rms

100 Ω1 Fµ



Vin

10 V dc

0

Vout

0 V dc10 V dc 100 Ω1 Fµ



ƒ = 100 Hz

0.63 V rms10 V rms

100 Ω1 Fµ

ƒ = 1 kHz

5.32 V rms10 V rms

100 Ω1 Fµ

ƒ = 10 kHz

9.87 V rms10 V rms

100 Ω1 Fµ



Vin

10 V dc

0

Vout

0 V dc10 V dc 100 Ω1 Fµ



Plotting the response:

ƒ = 100 Hz

0.63 V rms10 V rms

100 Ω1 Fµ

ƒ = 1 kHz

5.32 V rms10 V rms

100 Ω1 Fµ

ƒ = 10 kHz

9.87 V rms10 V rms

100 Ω1 Fµ

Vout (V)

f (kHz)

9.87

5.32

0.6300.01 0.1 1

10

98

7

6

5

4

3

21

10


Frequência de corte

Cutoff frequency (fc): The frequency at which the output voltage of a filter is 70.7% (1/√2) of the maximum output voltage or the frequency at which the output power is reduced to half it its maximum valu e.

Frequency response: In electric circuits, the variation of the output voltage (or current) over a specified range of frequencies.

fc

(10/√2)

Vout (V)

f (kHz)

9.87

5.32

0.6300.01 0.1 1

10

98

7

6

5

4

3

21

10

fc~1592 Hz


Função de Transferência de Circuitos RC: filtro RC passa-alto

Nesta montagem, a amplitude da tensão aos terminais da resistência (tensão de saída VOUT),

decresce à medida que a frequência do sinal de entrada, VIN, diminui.

A frequência de cortedeste circuito,ωc, é |H(ωc)|=|VOUT/V IN|=1/√2: ω=ωc=1/RC. A frequência

ωci=1/RCe ωcs=∞. A largura de banda éLB= ∞, comfci=1/2πRC.

O circuito comporta-se como umfiltro passa-alto: só os sinais de entrada com frequência superior

a ωc são transferidos,de forma eficiente, para a saída.

i

CVIN R VOUT

)/1(tan

)(1)(H)(

ZZ

Z

V

V)(H)(H

1

2

IN

OUTj

RC

RC

RCH

eCR

R

ω=θ

ω+

ω=ω=ω

+==ω=ω

−

θIOUT=0

Circuito RC Passa-Alto


Diagramas de Bode assimptóticos


Região opaca

g P = pq(P)1`(P) =1

1 1

1 KP

rsts

3rXuvw<uR

sts

P


K=1




122JF/CESDig 2018/2019

Filtros de ordem n


Filtros de ordem n

• A ordem n de um filtro é dada pelo número n de componente do circuito que

dependem da frequência;

• Atenuação à frequência de corte é igual a –n×3 dB, onde n é a ordem do filtro;

• Taxa de atenuação na região de opacidade: –n×20 dB/década

• A diferença de fase (uma década antes e uma década depois da frequência de

corte): −x ∙ 450 dB/década;

Dimensionamento de filtros• Dimensionamento das impedâncias:

• → ∙ C → / d → ∙ dA frequência de corte é invariante a esta transformação

• Dimensionamento da frequência: P′( = ∙ P(• → C → ∙ d → d/



124JF/CESDig 2018/2019

Resposta temporal de circuitos RC


Resposta de um circuito RC a pulsos de tensão

125

An RC integrator is a circuit that approximates the mathematical process of

integration. Integration is a summing process, and a basic integrator can

produce an output that is a running sum of the input under certain conditions.

A basic RC integrator circuit is simply a capacitor in series with a resistor and the source. The output is taken across the capacitor.

VS

R

C Vout



126

When a pulse generator is connected to the input of an RC integrator, the

capacitor will charge and discharge in response to the pulses.

When the input pulse goes HIGH, the pulse generator acts like a battery in series with a switch and the capacitor charges.

Switch closes

The output is an exponentially rising curve.

R

C



127

When the pulse generator goes low, the small internal impedance of the

generator makes it look like a closed switch has replaced the battery.

The output is an exponentially falling curve.

R

C

The pulse generator now acts like a closed switch and the capacitor discharges.


Carga e descarga de um condensador

The same shape curves are seen if a square

wave is used for the source.

VS

VC

VR

C

R

VS

What is the shape of the current curve?

The current has the same shape as VR.

128

τ RC=


Circuitos RC integrador e diferenciador

129

Integrator: A circuit producing an output that approaches the mathematical integral of the input.

Differentiator: A circuit producing an output that approaches the mathematical derivative of the input.


Circuito RC integrador

Waveforms for the RC integrator depend on the time constant (t) of the circuit.

R

C

What is τ if R = 10 kΩ and C = 0.022 µF? 220 µs

The output will reach steady state in about 5τ

Time constant

Transient time

A fixed time interval, set by R and C, or R and L values, that determines the time response of a circuit.

An interval equal to approximately five time constants.

If the time constant is short compared to the period of the input pulses, the capacitor will fully charge and discharge.

For an RC circuit, = ~. The output will reach 63% of the final value in 1 .


Circuito RC integrador

Se o período da onda de entrada diminuir (i.e., a frequência aumentar), a onda de saída tende para o valor médio da onda de entrada, e onda de saída vai-se assemelhando cada vez mais a uma onda triangular, mas com uma amplitude menor.

A onda triangular pode ser interpretada como o “integral” de uma onda quadrada.

t

t

t

t

Vin

Vout

Vout

Vout


Resposta de um circuito CR a pulsos de tensão

An RC differentiator is a circuit that approximates the mathematical process of

differentiation. Differentiation is a process that finds the rate of change, and a

basic differentiator can produce an output that is the rate of change of the

input under certain conditions.

A basic RC differentiator circuit is

simply a resistor in series with a

capacitor and the source. The output

is taken across the resistor.

VS R

C

Vout



When a pulse generator is connected to the input of an RC differentiator, the

capacitor appears as an instantaneous short to the rising edge and passes it

to the resistor.

The capacitor looks like a

short to the rising edge

because voltage across

C cannot change

instantaneously.

During this first instant, the output follows the input.

0

VC = 0



After the initial edge has passed, the capacitor charges and the output

voltage decreases exponentially.

The voltage across C is the traditional charging waveform.

The output decreases as the pulse levels off.



The falling edge is a rapid change, so it is passed to the output because the

capacitor voltage cannot change instantaneously. The type of response shown

happens when τ is much less than the pulse width (τ<< tw).

The voltage across C when the input goes low decreases exponentially.

After dropping to a negative value, the output voltage increases exponentially as the capacitor discharges.



If τ is long compared to the pulse width, the output does have time to return to the original baseline before the pulse ends. The resulting output looks like a pulse with “droop”.

Vin

5τ = tw

5τ >> tw

tw

When 5τ = tw, the pulse has just returned to the baseline when it repeats.

The output shape is dependent on the ratio of τ to tw.


Circuito CR diferenciador

The falling edge is a rapid change, so it is passed to the output because the

capacitor voltage cannot change instantaneously. The type of response shown

happens when τ is much less than the pulse width (τ<< tw).

The voltage across C when the input goes low decreases exponentially.

After dropping to a negative value, the output voltage increases exponentially as the capacitor discharges.


Aplicações de circuitos com condensadores

138


Filtragem de sinais usando condensadores(Power supply filtering )

There are many applications for capacitors. One is in filters, such as the power supply filter shown here.

Rectifier50 Hz ac

CLoad resistance

The filter smoothes the pulsating dc from the rectifier.

139


Acoplamento/desacoplamento de sinais dc e ac usando condensadores

(Coupling capacitors)

Coupling capacitors are used to pass an ac signal from one stage to another while blocking dc.

The capacitor isolates dc between the amplifier stages, preventing dc in one stage from affecting the other stage.

0 V 0 V

3 V

Amplifierstage 1

Amplifierstage 2

R2

R1C

+V

Input Output

140


Bypass capacitors

Another application is to bypass an ac signal to ground but retain a dc value. This is widely done to affect gain in amplifiers.

The bypass capacitor places point A at ac ground, keeping only a dc value at point A.

R2

R1

C

0 V

dc plusac

Point in circuit whereonly dc is required

0 V

dc only

A

141


Aplicação do circuito RC integrador

An application of an integrator is to generate a time delay. The voltage at B

rises as the capacitor charges until the threshold circuit detects that the

capacitor has reached a predetermined level.

SW closes

Threshold

Time delay

R VoutVin A B VA

VB

Vout

Threshold circuitSW

C


Representação de geradores e instrumentos de medida

• Modelos de geradores de tensão e de corrente

• Modelos de voltímetro e de amperímetro

• Resistências/impedâncias (interna) do voltímetro e do amperímetro

• Impedância do osciloscópio


Modelos dos geradores de tensão e de corrente

144144

+-

+ v0–

R0V i0

+ v=v0+i0R0V

– +

-

i0 R0I

i=i0-i’

+ v=i’R0I

–

i’

+-

+ v0–

R1=100 kΩ i

+ vmedido

–

10 V VRV

+-

+ v0–

R1=100 kΩ imedido

10 V

A

RA

Modelos do voltímetro e do amperímetro

Fonte de tensão Fonte de corrente

Fonte de tensão ideal Fonte de corrente ideal

R0V: Resistência interna da fonte R0I: Resistência interna da fonte

vmedido=v0RV/(R1+RV) imedido=v0/(R1+RA)

Voltímetro Amperímetro

Voltímetro ideal Amperímetro ideal


Impedância do osciloscópioO osciloscópio é um instrumento de medida de sinais elétricos/eletrónicos que apresenta representações gráficas duas dimensões de um ou mais sinais elétricos (de acordo com a quantidade de canais de entrada). O eixo vertical (y) do ecrã representa a magnitude do sinal (tensão) e o eixo horizontal (x) representa o tempo, tornando o instrumento útil para visualizar sinais periódicos ou sinais variáveis no tempo. https://pt.wikipedia.org/wiki/Oscilosc%C3%B3pio

Cada uma das entradas do osciloscópio possui uma impedância, normalmente uma resistência em paralelo com um condensador. A impedância de um osciloscópio deve ser a maior possível. Os valores típicos da resistência e da capacidade são da ordem de 1 MΩ e de 10 a 80 pF, respectivamente.

“The input impedance is made a specific nominal value, rather than arbitrarily high, because of the common use of X10 probes. With a known input impedance to the oscilloscope, the probe designer can ensure that the probe input impedance is exactly ten times this figure (actually oscilloscope plus probe cable impedance). Since the impedance included the input capacitance and the probe is an impedance divider circuit, the result is that the waveform being measured is not distorted by the RC circuit formed by the probe resistance and the capacitance of the input (or the cable capacitance which is generally higher).” https://en.wikipedia.org/wiki/Nominal_impedance#Oscilloscopes

Usando os dois canais do osciloscópio é possível observar no canal 1 o sinal aplicado ao circuito e no canal 2 a resposta do circuito ao sinal aplicado.


Impedâncias características “padrão”

146

Após o advento do radar [técnica de localização e determinação de distância de um objeto afastado (avião, submarino etc.) por meio da emissão de ondas radioelétricas e a detecção e análise do pulso refletido pelo objeto], a industria sentiu necessidade de definir uma “impedância padrão” para os sistemas de transmissão, em particular para os cabos coaxiais usados para conduzir os sinais. O padrão mais usado corresponde à impedância de 50 Ω, e este valor resulta da satisfação do compromisso entre dois requisitos: i) perdas de transmissão mínimas; ii) capacidade de transmitir a maior potencia possível.

Atualmente, os sistema de medição e caracterização mais avançados em electrónica e optoelectrónica empregam componentes (equipamentos, cabos, conectores, circuitos, etc.) cuja impedâncias de entrada (e de saída) e/ou característica é 50 Ω. Alguns dos osciloscópio e geradores de sinais mais comuns permitem definir as impedâncias de entrada e de saída, respectivamente, em função da análise que se pretende fazer. Antes de começar a utilizar um dado equipamento devemos verificar qual é a impedância que está a ser considerada pelo equipamento.

No campo da radiofrequência, das micro-ondas e das ondas milimétricas os sistemas são desenhados, quase sem excepção com impedâncias de entrada e de saída (sempre que for aplicável) de 50 Ω, e transmissão guiada de sinais é feita usando linhas de transmissão (cabos coaxiais, linhas “micro-strip”, linhas “coplanar waveguide (CPW)”, etc.) com impedância característica de 50 Ω.

Impedância dos cabos de vídeo: Os sinais de vídeo são geralmente transmitidos usando cabos coaxiais com impedância igual a 75 Ω, fazendo com que 75 Ω se tenha tornado um padrão quase universal para os cabos coaxiais para vídeo.

Impedância de altifalantes e colunas: As impedâncias de alto-falante são mantidas relativamente baixas em comparação com outros componentes de áudio, de modo que a potência de áudio necessária pode ser transmitida sem o uso inconveniente (e perigoso) de altas tensões. A impedância nominal mais comum para alto-falantes é de 8 Ω.



JF/CESDig 2018/2019


• Bobines/Indutâncias

• Associação de indutâncias

• Impediência e reatância indutiva

• Circuitos RL

• Resposta temporal

• Resposta em frequência

• Filtros RL e RCL

• Filtros passa-baixo, passa-alto, passa-banda, rejeita-banda

• Filtros de ordem superior (filtros de ordem n)

• Taxa de atenuação por oitava e por década

12-10-2018 147


When a length of wire is formed into a coil., it becomes a basic inductor. When there is current in the inductor, a three-dimensional magnetic field is created.

A change in current causes the magnetic field to change. This in turn induces a voltage across the inductor that opposes the original change in current.

NS

Bobine/indutor básico

Uma bobine/indutância é um dispositivo elétrico que produz um fluxo magnético quando percorrida por uma corrente elétrica. Se a corrente for varável o fluxo criado também é variável, e a variação de fluxo induz uma força eletromotriz de autoindução na própria bobine.

A grandeza que caracteriza uma indutância é o seu coeficiente de autoindução, que se designa simplesmente por indutância e que se representa por L.

Inductance (L): The property of an inductor whereby a change in current causes the inductor to produce a voltage that opposes the change in current.

148


Bobines e indutância

NS

Lei de Faraday: The amount of voltage induced in a coil is directly proportional to the rate of change of the magnetic field with respect to the coil.

Lei de Lenz: When the current through a coil changes and an induced voltage is created as a result of the changing magnetic field, the direction of the induced voltage is such that it always opposes the change in the current.

A tensão aos terminais de uma bobine é proporcional à taxa temporal de variação da corrente que atravessa a bobine:

= d ()

1 henry corresponde à indutância de uma bobine quando a corrente, a variar à taxa de 1 A por segundo, induz uma diferença de tensão de 1 V aos terminais da bobine.

A constante de proporcionalidade chama-se indutância e representa-se por L. A unidade de indutância é o henry (H):

() = d 1()& = = d ' =

d ∙ () energia armazenada num bobine

149


Fatores que afetam a indutânciaFour factors affect the amount of inductance for a coil. The equation for the inductance of a coil is

2N AL

l

µ=where

L = inductance in HenryN = number of turns of wirem = permeability in H/m (same as Wb/At-m) l = coil length on meters

What is the inductance of a 2 cm long, 150 turn coil wrapped on an low carbon steel core that is 0.5 cm diameter? The permeability of low carbon steel is 2.5 x10-4 H/m (Wb/At-m).

( )22 5 2π π 0.0025 m 7.85 10 mA r −= = = ×2N A

Ll

µ=( ) ( )( )2 4 5 2150 t 2.5 10 Wb/At-m 7.85 10 m

0.02 m

− −× ×=

= 22 mH

150

Símbolo da bobine/indutância


Modelo de uma bobine/indutor real

In addition to inductance, actual inductors have winding resistance (RW) due to the

resistance of the wire and winding capacitance (CW) between turns. An equivalent

circuit for a practical inductor including these effects is shown:

LRW

CW

Notice that the winding resistance is in series with the coil and the winding

capacitance is in parallel with both.

151

Símbolo da bobine/indutância ideal

Indutância real


Circuito simples para demonstrar a lei de Lenz

Inicialmente o interruptor SW está aberto, e a bobine L e a resistência R1 são percorridas por uma pequena corrente.

−R1

SW

R2VS

L

+

− +

Imediatamente após se fechar o interruptor SW aparece uma tensão aos terminais de L que tende a a opor-se a qualquer alteração na corrente.

−

−

R1

SW

R2VS

L

+

+

− +

E o amperímetro indica exactamente a mesma intensidade de corrente que registava antes de fecharmos o interruptor SW.

+ VL=0 -

152


Circuito simples para demonstrar a lei de Lenz

Algum tempo depois (5@ = 5d/(1//2)), a corrente estabiliza num valor mais elevado, uma vez que tensão aos terminais da bobine decaiu novamente para zero.

−

−

R1

SW

R2VS

L

+

+

− +

Initially, the meter reads same current as before the switch was closed.

+−

−R1

SW

R2VS

L

+

O amperímetro indica um valor superior de corrente devido à alteração da carga.

Qual foi a variação da carga?

+ VL=0 -

153


Tipos de bobines/indutores

Common symbols for inductors (coils) are

Air core Iron core Ferrite core Variable

There are a variety of inductors, depending on the amount of inductance

required and the application. Some, with fine wires, are encapsulated and may

appear like a resistor.

154


Bobines comerciais

Inductors come in a variety of sizes. A few common ones are shown here.

Encapsulated Torroid coil Variable

155


Associação de bobines

156


Associação de indutores em série

When inductors are connected in series, the total inductance is the sum of the individual inductors. The general equation for inductors in series is

2.18 mH

T 1 2 3 ... nL L L L L= + + +

If a 1.5 mH inductor is connected in series with an 680 µH inductor, the total inductance is

L1 L2

1.5 mH 680 Hµ

157


Associação de indutores em paraleloWhen inductors are connected in parallel, the total inductance is smaller than the smallest one. The general equation for inductors in parallel is

The total inductance of two inductors is

…or you can use the product-over-sum rule.

T

1 2 3 T

11 1 1 1

...L

L L L L

=+ + + +

T

1 2

11 1

L

L L

=+

If a 1.5 mH inductor is connected in parallel with an 680 µH inductor, the total inductance is

468 µH

L1 L2

1.5 mH 680 Hµ

158


Impedância de uma bobine

159


Impedância de uma bobine

D = K2LMd KPd→ 6KPd

VL 0

0

90°

I

1`3XY→

+

XY→pq KPd ∙

:;< d

3XY→ d

1

KPd ∙

When a sine wave is applied to an inductor there is a phase shift (desfasamento) between voltage and current such that voltage always leads the current by 90o.

I.e., a VL está adiantada 900 em relação à IL.

160


Reactância indutivaInductive reactance is the opposition to ac by an inductor. The equation for inductive reactance is

The reactance of a 33 µH inductor when a frequency of 550 kHz is applied is 114 Ω

2πLX fL=

When inductors are in series, the total reactance is the sum of the individual reactances. That is,

Assume three 220 µH inductors are in series with a 455 kHz ac source. What is the total reactance?

1.89 kΩ

L( ) L1 L2 L3 Ltot nX X X X X= + + + ⋅⋅⋅+

The reactance of each inductor is

( )( )L 2 2 455 kHz 220 µH 629 X fLπ π= = = Ω

L( ) L1 L2 L3

629 629 629totX X X X= + +

= Ω + Ω + Ω =

E = PdD = K2LMd = KE

161


Inductive reactance

When inductors are in parallel, the total reactance is the reciprocal of the sum of the reciprocals of the individual reactances. That is,

If the three 220 µH inductors from the last example are placed in parallel with the 455 kHz ac source, what is the total reactance?

210 Ω

L( )

L1 L2 L3 L

11 1 1 1tot

n

X

X X X X

=+ + + ⋅⋅⋅ +

The reactance of each inductor is 629 Ω

L( )

L1 L2 L3

1 11 1 1 1 1 1

+ +629 629 629

totX

X X X

= = =+ +

Ω Ω Ω

162


Resposta temporal de circuitos com bobines

163


Bobines em circuitos DC

When an inductor is connected in series with a resistor and dc source, the current change is exponential.

R

L

t0Current after switch closure

t0 Inductor voltage after switch closure

Vinitial

Ifinal

= d 1() → = + !"

A corrente numa bobine não pode variar bruscamente.

= constante → curto − circuito τL

R=

@ = d/

0.63Ifinal

0.37Vinicial

= Z9 1 − 3!/?

164


Bobines em circuitos DC

The same shape curves are seen if a square wave is used for the source.

VSR

LVS

τL

R=

= d ()

#

VL

VR

=Constante de tempo de um circuito RL:

= 9 + = () + d ()Z9 = () +

91() → = Z

9 1 − 3!/(/9)

= 3!/(/9)165


Circuito RL integrador

166


Like the RC integrator, an RL integrator is a circuit that approximates the mathematical process of integration. Under equivalent conditions, the waveforms look like the RCintegrator. For an RLcircuit, τ = L/R.

A basic RL integrator circuit is a resistor in series with an inductor and the source. The output is taken across the resistor. VS R

L

Vout

What is the time constant if R = 22 kΩand L = 22 µH? 1.0 ms


167


When the pulse generator output goes high, a voltage immediately appears across the inductor in accordance with Lenz’s law. The instantaneous current is zero, so the resistor voltage is initially zero.

The output is initially zero because there is no current.

VS

R

L

+ −

The induced voltage across L opposes the initial rise of the pulse.

0 V



168


At the top of the input pulse, the inductor voltage decreases exponentially and current increases. As a result, the voltage across the resistor increases exponentially. As in the case of the RC integrator, the output will be 63% of the final value in 1τ.

The output voltage increases as current builds in the circuit.

VS

R

L

+ −

The induced voltage across L decreases.



169


When the pulse goes low, a reverse voltage is induced across L opposing

the change. The inductor voltage initially is a negative voltage that is equal

and opposite to the generator; then it exponentially increases.

The output voltage decreases as the magnetic field around L collapses.

VS

R

L

+ −

The induced voltage across Linitially opposes the change in the source voltage.

Note that these waveforms were the same in the RC integrator.



170


Circuito RL diferenciador

171


An RLdifferentiator is also a circuit that approximates the mathematical process of differentiation. It can produce an output that is the rate of change of the input under certain conditions.

A basic RLdifferentiator circuit is an inductor in series with a resistor and the source. The output is taken across the inductor.

VS L

R

Vout


172


When a pulse generator is connected to the input of an RLdifferentiator, the inductor has a voltage induced across it that opposes the source; initially, no current is in the circuit.

Current is initially zero, so VR= 0.

During this first instant, the inductor develops a voltage equal and opposite to the source voltage.

VR = 0

VS

L

R

+

−



173


After the initial edge has passed, current increases in the circuit. Eventually, the current reaches a steady state value given by Ohm’s law.

The voltage across R increases as current increases.

The output decreases as the pulse levels off.

VS

L

R

+

−



174


Next, the falling edge of the pulse causes a (negative) voltage to be induced across the inductor that opposes the change. The current decreases as the magnetic field collapses.

The voltage across R decreases as current decreases.

The output decreases initially and then increases exponentially.

VS

L

R

+

−



175


If τ is long compared to the pulse width, the output looks like a pulse with “droop”.

Vin

5τ = tw

5τ >> tw

twWhen 5τ = tw, the pulse has just returned to the baseline when it repeats.

As in the case of the RCdifferentiator, the output shape is dependent on the ratio of τ to tw.


176


Sinusoidal response of RL circuits

When both resistance and inductance are in a series circuit, the phase angle (desfasamento) between the applied voltage and total current is between 0° and 90°, depending on the values of resistance and reactance.

I

LR

VR

VR lags VS VL leads VS

I lags VS

VS

VL

Desfasamento entre dois sinais: diferença de fase e ntre esses sinais.

177


Resposta em frequência de circuitos RL

RL time constant: A fixed time interval set by the L and R values, that determines the time response of a circuit. It equals the ratio of L/R.

178


φ (phase lag)φ

R

VL

Vout

L

VoutVin

VinVout

Vin

Reversing the components in the previous circuit produces a circuit that is a

basic lag network (a tensão de saída está atrasada relativamente à tensão da

entrada). This circuit is also a basic low-pass filter, a circuit that passes low

frequencies and rejects all others.

Desfasamento (atraso) em circuito LR

179


Frequency Response of RL Circuits

Series RLcircuits have a frequency response similar to series RCcircuits. In the case of the low-pass response shown here, the output is taken across the resistor.

100 Ω10 mH

10 V dc

VoutVin

10 V dc

0

10 V dc

0

180



100 Ω10 mH

10 V dc

VoutVin

10 V dc

0

10 V dc

0100 Ωƒ = 1 kHz

8.46 V rms10 V rms 10 mH


181



100 Ω10 mH

10 V dc

VoutVin

10 V dc

0

10 V dc

0100 Ωƒ = 1 kHz


100 Ω10 mH

ƒ = 10 kHz

1.57 V rms

10 V rms


182



100 Ω10 mH

10 V dc

VoutVin

10 V dc

0

10 V dc

0100 Ωƒ = 1 kHz


100 Ω10 mH

ƒ = 10 kHz

1.57 V rms

10 V rms

100 Ω10 mH

ƒ = 20 kHz

0.79 V rms

10 V rms


183



100 Ω10 mH

10 V dc

VoutVin

10 V dc

0

10 V dc

0


100 Ωƒ = 1 kHz


100 Ω10 mH

ƒ = 10 kHz

1.57 V rms

10 V rms

100 Ω10 mH

ƒ = 20 kHz

0.79 V rms

10 V rms

Vout (V)

9.98

8.46

1.570.79

0.1 1 10 20 100f (kHz)

9

8

7

6

5

4

3

2

1


Cutoff frequency (fc): The frequency at which the output voltage of a filter is 70.7% (1/√2) of the maximum output voltage.

fc~1592 Hz

184


Resposta em frequência de circuitos RL

RL time constant: A fixed time interval set by the L and R values, that determines the time response of a circuit. It equals the ratio of L/R.

185


θ

(phase lead)

φ

φ

R

VR

Vout

L VoutVin

Vin

Vout Vin

For a given frequency, a series RL circuit can be used to produce a phase lead

by a specific amount between an input voltage and an output by taking the

output across the inductor (adiantamento da tensão de saída relativamente à

tensão da entrada). This circuit is also a basic high-pass filter, a circuit that

passes high frequencies and rejects all others.

Desfasamento (adiantamento) em circuitos RL

186



Reversing the position of the Rand L components, produces the high-pass response. The output is taken across the inductor.

Vin

10 V dc

0

Vout

0 V dc10 V dc100 Ω

10 mH

187



Vin

10 V dc

0

Vout

0 V dc10 V dc100 Ω

10 mHƒ = 100 Hz0.63 V rms

10 V rms 10010 mH


188



Vin

10 V dc

0

Vout

0 V dc10 V dc100 Ω

10 mHƒ = 100 Hz0.63 V rms

10 V rms 10010 mHƒ = 1 kHz

5.32 V rms10 V rms

100 Ω10 mH


189



Vin

10 V dc

0

Vout

0 V dc10 V dc100 Ω

10 mHƒ = 100 Hz0.63 V rms

10 V rms 10010 mHƒ = 1 kHz

5.32 V rms10 V rms

100 Ω10 mH

ƒ = 10 kHz

9.87 V rms10 V rms

100 Ω10 mH


190




Vin

10 V dc

0

Vout

0 V dc10 V dc100 Ω

10 mH


Vout (V)

f (kHz)

9.87

5.32

0.6300.01 0.1 1

10

98

7

6

5

4

3

21

10

ƒ = 100 Hz0.63 V rms

10 V rms 10010 mHƒ = 1 kHz

5.32 V rms10 V rms

100 Ω10 mH

ƒ = 10 kHz

9.87 V rms10 V rms

100 Ω10 mH

Cutoff frequency (fc): The frequency at which the output voltage of a filter is 70.7% (1/√2) of the maximum output voltage.

fc~1592 Hz

191


Potência em Corrente Alternada

192


Potência em Corrente Alternada

Seja um elemento de um circuito percorrido por uma correntei(t)=Ipcos(ωt), e aosterminais do qual se estabelece uma tensãov(t)=Vpcos(ωt+φ).

A potência instantâneafornecida ao elemento é dada por:

p=v.i=[Vpcos(ωt+φ)][ Ipcos(ωt)].

Valores positivos de potência indicam que o elemento está a dissipar/armazenar

energia; valores negativos indicam que o elemento está a “gerar”/devolver energia

ao circuito. Em corrente alternada definem-se as seguinte potências:

Potência Aparente, pZ: potência que é transferida pela fonte ao circuito,

pZ=i2Z (unidade: volt-ampere, VA)

Potência Reactiva, pX: potência fornecida aos elementos reativos

pX=i2X (unidade: volt-ampere reactivo, VAr)

193


Potência em Corrente AlternadaPotência Real (verdadeira), pR: parte da potência dissipada no circuito

pR=i2R (unidade: watt, W).

A potência média é dada por: P=<p>=1/2VpIpcos(φ)=VEfIEfcos(φ).

O factor cos(φ) é designadofactor de potência.Em circuitos de potência é desejável que cos(φ) ~1. Caso contrário, para um dado

par tensão e potência, é necessário fornecer ao circuito umaintensidade de

corrente superior ao que ser necessário se o fator de potência fosse igual a 1, o

que origina perdas elevadas, por exemplo, nas linhas de transmissão.

Exercício de aplicação:Num secador de cabelo está indicado 1500 W a 220 V. Determine a resistência,a corrente

eficaz e a potência instantânea máxima. Assuma que o secador é uma resistência pura.Sol.:32,3Ω, 6,8 A, 3000 W. (Alguns vendedores de equipamentos de áudio, e não só, anunciam

valores de potência que correspondem aos valores máximos de potência como sendo a

potência média, o que é não correto).

194


Potência num indutor

True Power : Ideally, inductors do not dissipate power. However, a small amount of power is dissipated in winding resistance given by the equation:

Ptrue = (Irms)2RW

Reactive Power :

Reactive power is a measure of the rate at which the inductor stores and returns energy. One form of the reactive power equation is:

Pr=VrmsIrms

The unit for reactive power is the VAR.

195


Fator de qualidade de uma bobine

An equivalent circuit for a practical inductor includes the winding resistance (RW)

due to the resistance of the wire and winding capacitance (CW) between turns:

LRW

CW

Notice that the winding resistance is in series with the

coil and the winding capacitance is in parallel with both.

The quality factor (Q) of a coil is given by the ratio of reactive power to true power.

2

2L

W

I XQ

I R= For a series circuit, I cancels, leaving:

L

W

XQ

R=



JF/CESDig 2018/2019


• Circuitos RCL série

• Impedância

• Ressonância

• Filtros

• Filtros não ideais

• Circuitos RCL paralelo

• Impedância

• Ressonância

• Filtros

• Filtros não ideais

12-10-2018 197


Circuito RLC série

VS

R L C

198


Circuito RLC série

VS

R L C

When a circuit contains an inductor and capacitor in series, the reactance of each tend to cancel. The total reactance is given by

tot L CX X X= −

The total impedance is given by 2 2tot totZ R X= +

The phase angle is given by 1tan totX

Rθ − =

199


Rea

ctan

cef

XC XL

In a series RLCcircuit, the circuit can be capacitive or inductive, depending on the frequency.

At the frequency where XC=XL, the circuit is at series resonance.

Series resonance

XC=XL

Below the resonant frequency, the circuit is predominantly capacitive.

Above the resonant frequency, the circuit is predominantly inductive.

XC>XL XL>XC

Circuito RLC série: variação de XL e XC com a frequência

Frequência de ressonância

200


VS

R L C

What is the total impedance and phase angle of the series RLCcircuit if R= 1.0 kΩ, XL = 2.0 kΩ, and XC = 5.0 kΩ?

1.0 kΩ XC = 5.0 kΩ

The total reactance is 2.0 k 5.0 k 3.0 ktot L CX X X= − = Ω − Ω = Ω

The total impedance is2 2 2 21.0 k +3.0 ktot totZ R X= + = Ω Ω = 3.16 kΩ

XL = 2.0 kΩ

The circuit is capacitive, so I leads V by 71.6o.

The phase angle is 1 1 3.0 ktan tan

1.0 ktotX

Rθ − − Ω = = = Ω

71.6o

Circuito RLC série: impedância

201


What is the magnitude of the impedance for the circuit?

R L C

VS 330 µH

f = 100 kHz

470 Ω 2000 pF

( ) ( )2 2 100 kHz 330 H 207 LX fLπ π µ= = = Ω

( )( )1 1

796 2 2 100 kHz 2000 pFCX

fCπ π= = = Ω

207 796 589 tot L CX X X= − = Ω − Ω = Ω

( ) ( )2 2 = 470 589 Z Ω + Ω = 753 Ω


202


R L C

VS

What is the total impedance for the circuit when the frequency is increased to 400 Hz?

330 µHf = 400 kHz470 Ω 2000 pF

( )( )2 2 400 kHz 330 H 829 LX fLπ π µ= = = Ω

( )( )1 1

199 2 2 400 kHz 2000 pFCX

fCπ π= = = Ω

The circuit is now inductive.

829 199 630 tot L CX X X= − = Ω − Ω = Ω

( ) ( )2 2 = 470 630 Z Ω + Ω = 786 Ω


203


The voltages across the RLCcomponents must add to the source voltage in accordance with KVL. Because of the opposite phase shift due to L and C, VL and VC effectively subtract.

0

Notice thatVC is out of phase with VL. When they are algebraically added, the result is….

VC

VL

This example is inductive.

Tensão em circuito RLC série

VS

R L C

204


At series resonance, XC and XL cancel. VC and VL also cancel because the voltages are equal and opposite. The circuit is purely resistive at resonance.

0

Algebraic sum is zero.

Circuito RLC série: ressonância

VS

R L C

205


The formula for resonance can be found by setting XC = XL. The result is

1

2rf

LCπ=

What is the resonant frequency for the circuit?

R L C

VS

330 µH470 Ω 2000 pF

( )( )

1

21

2 330 µH 2000 pF

rfLCπ

π

=

=

= 196 kHz


206


What is VR at resonance?

Ideally, at resonance the sum of VL and VC is zero.

R L C

VS

330 µH470 Ω 2000 pF

5.0 Vrms

5.0 Vrms

V = 0VS

By KVL, VR = VS

5.0 Vrms


207


XL

f

XC

X

The general shape of the impedance versus frequency for a series RLC circuit is superimposed on the curves for XL and XC. Notice that at the resonant frequency, the circuit is resistive, and Z = R.

Z

Series resonance

Z = R

Circuito RLC série: impedância vs frequência

208


Summary of important concepts for series resonance:

• Capacitive and inductive reactances are equal.

• Total impedance is a minimum and is resistive.

• The current is maximum.

• The phase angle between VS and IS is zero.

• fr is given by 1

2rf

LCπ=


209


An application of series resonant circuits is in filters. A band-pass filter allows signals within a range of frequencies to pass.

R

L CVoutVin

Resonant circuit

f

Vout

Series resonance

Circuit response:

Circuito RLC série: filtro ressonante

210


An application of series resonant circuits is in filters. A band-pass filterallows signals within a range of frequencies to pass. The response has a peak because at the seriesresonant frequency, the current is maximum at resonance and falls off before and after resonance. This develops the maximum voltage across the resistor at resonance. The bandwidth (BW) of the filter is the range of frequencies for which the output is equal to or greater than 70.7% of the maximum value. f1 and f2 are commonly referred to as the critical frequencies, cutoff frequenciesor half-power frequencies.

0.707

f1 fr f2

BW

f

I or VoutPassband

1.0


R

L CVoutVin

Resonant circuitCircuit response:

211


Filter responses are often given in terms of decibels, which is defined as

dB 10log out

in

P

P

=

dB 20log out

in

V

V

=

Another useful definition for the decibel, when measuring voltages across the same impedance is

Because it is a ratio, the decibel is dimensionless. One of the most important decibel ratios occurs when the power ratio is 1:2. This is called the −3 dB frequency, because

1dB 10log 3 dB

2 = = −

Decibéis

212


Fator de qualidade de uma bobine

The quality factor (Q) of a coil is given by the ratio of reactive power to true power:

2

2L

W

I XQ

I R=

For a series circuit, I cancels, leaving:

L

W

XQ

R=

213


Selectivity describes the basic frequency response of a resonant circuit. (The −3 dB frequencies are marked by the dots.)

f0

BW3

Least Selectivity

BW2

Medium Selectivity

BW1

Greatest Selectivity

The bandwidth is inversely proportional to Q in accordance with the formula,

rfBWQ

=

Which curve represents the highest Q?

The one with the greatest selectivity.

Circuito RLC série: seletividade

214


By taking the output across the resonant circuit, a band-stop (or notch) filter is produced.

RVoutVin

f

Vout

Circuit response:

L

C

Resonant circuit

f1 fr

f2

BW

Stopband

0.707

1

f2


215


Circuito RLC paralelo

R L CVS

216


The currents in the RLCcomponents must add to the source current in accordance with KCL. Because of the opposite phase shift due to L and C, IL

and IC effectively subtract.

Notice thatIC is out of phase with IL. When they are algebraically added, the result is….

IC

IL

0

Corrente em circuitos RLC paralelo

217


0

Ideally, at parallel resonance, IC and IL cancel because the currents are equal and opposite. The circuit is purely resistive at resonance.

The algebraic sum is zero.

Notice thatIC is out of phase with IL. When they are algebraically added, the result is….

IC

IL

Circuito RLC paralelo: ressonância

218


In practical circuits, when the coil resistance is considered, there is a small

current at resonance and the resonant frequency is not exactly given by the ideal

equation. The Q of the coil affects the equation for resonance:

2

2

1

12r

Qf

QLCπ=

+ (non-ideal)

For Q >10, the difference between the ideal and the non-ideal formula is less than

1%, and generally can be ignored.

Circuito RLC paralelo: ressonância em circuitos não ideais

LRW

CW

219


At the parallel resonant frequency, impedance is maximum, so current is a minimum at resonance. The bandwidth (BW) can be defined in terms of the impedance curve.

f

Zmax

0.707Zmax

f1 fr f2

BW

Ztot

A parallel resonant circuit is commonly referred to as a tank circuit because of its ability to store energy like a storage tank.

Circuito RLC paralelo: largura de banda

R LC

VS

220


Summary of important concepts for parallel resonance:

• Capacitive and inductive susceptance are equal.

• Total impedance is a maximum (ideally infinite).

• The current is minimum.

• The phase angle between VS and IS is zero.

• fr is given by 1

2rf

LCπ=

Circuito RLC paralelo: ressonância

221


Parallel resonant circuits can also be used for band-pass or band-stop filters. A basic band-pass filter is shown.

R

VinVout

Parallel resonant band-pass filter

CLResonant circuit

0.707

f1 fr f2

BW

f

VoutPassband

Circuit response:

1.0

Circuito RLC paralelo: filtro passa-banda

222


For the band-stop filter, the resonant circuit and resistance are reversed as shown here.

f

Vout

f1 fr

BW

Stopband

0.707

1

R

Vin Vout

Parallel resonant band-stop filter

C

L

Resonant circuit

Circuit response:

f2

Circuito RLC paralelo: filtro rejeita-banda

223



224JF/CESDig 2018/2019

Filtros: conceitos e definições a reter


Filtros de ordem n

• A ordem n de um filtro é dada pelo número n de componente do circuito que

dependem da frequência;

• Atenuação à frequência de corte é igual a –n×3 dB, onde n é a ordem do filtro;

• Taxa de atenuação na região de opacidade: –n×20 dB/década

• A diferença de fase (uma década antes e uma década depois da frequência de

corte): −x ∙ 450 dB/década;

Dimensionamento de filtros• Dimensionamento das impedâncias:

• → ∙ C → / d → ∙ dA frequência de corte é invariante a esta transformação

• Dimensionamento da frequência: P′( = ∙ P(• → C → ∙ d → d/


• A band-pass filter allows frequencies between two critical frequencies and rejects all others.

• A band-stop filter rejects frequencies between two critical frequencies and passes all others.

• Band-pass and band-stop filters can be made from both series and parallel resonant circuits.

• The bandwidth of a resonant filter is determined by the Q and the resonant frequency.

• The output voltage at a critical frequency is 70.7% of the maximum.

Filtros ressonantes: ideias a reter

226


Series resonance:

Resonant frequency (fr):

Parallel resonance:

Tank circuit:

A condition in a series RLC circuit in which the reactances ideally cancel and the impedance is a minimum.

The frequency at which resonance occurs; also known as the center frequency.

A condition in a parallel RLC circuit in which the reactances ideally are equal and the impedance is a maximum.

A parallel resonant circuit.

Termos s reter

227


Half-power frequency:

Decibel:

Selectivity:

The frequency at which the output power of a resonant circuit is 50% of the maximum value (the output voltage is 70.7% of maximum); another name for critical or cutoff frequency.

Ten times the logarithmic ratio of two powers.

A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity.

Termos a reter

228


Diagramas de Bode assimptóticos: exemplo circuito CR passa-alto

229



l P = pq(P)1`(P) =1

1 1

1 KP

l P 3X∅

rsts

3rXuvw<uR

sts , P


Vamos considerar três situações: para o módulo da função de transferência

i) ω<< ωC ∶ gP →ωωC

, gPdB 20 Log 10ωωC

(-20 dB por década)

ii) ω= ωC ∶ gP

, g P dB 20 Log 10

63dB

iii) ω>> ωC∶ gP → 1, g P dB 20 Log 10 1 0 dB

Da mesma forma para a fase função de transferência: i) ω<< ωC ∶ ∅P → 900

ii) ω= ωC ∶ ∅P 450, iii) ω>> ωC∶ ∅P → 00,




Região opaca

K=1


http://www.ufrgs.br/eng04030/Aulas/teoria/cap_12/respfreq.htm

g P→∞ → 1

g P→0 → P/Pc

φ P→0 → 900

φ P→∞ → 00

φ P P 450

l P pqP

1`P

1

1 1

1 KP

l P 3X∅

1

1 PP

3rXuvw<uR

YtY

P


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